One thing you can try is to animate the surface emission, however that would mean that the water would start to appear from more than the object's face.  Instead you could have a hidden dup of your emission object that is the actual emission object, scaled down to your minimum flow.

[![enter image description here][1]][1][<img src="https://blend-exchange.com/embedImage.png?bid=qgg2BRRD" />](https://blend-exchange.com/b/qgg2BRRD/)

I'm still downloading 3.0, but probably similar in 2.93.5...

You could try a trick where you have a hidden (non-renderable) collision object that is the same shape as your emitter object, and you could vary its scale to have it interfere more or less.

[![enter image description here][2]][2][<img src="https://blend-exchange.com/embedImage.png?bid=jX46KsQn" />](https://blend-exchange.com/b/jX46KsQn/)

Another idea is to animate the particle radius:

[![enter image description here][3]][3]
[<img src="https://blend-exchange.com/embedImage.png?bid=DqLRWYoS" />](https://blend-exchange.com/b/DqLRWYoS/)

I finally got blender 3.0 installed.   The blender uploader doesn't seem to recognice 3.x file yet, so here is a [Dropbox link][4].


  [1]: https://i.sstatic.net/uw70q.png
  [2]: https://i.sstatic.net/9PcOS.png
  [3]: https://i.sstatic.net/OqvUS.png
  [4]: https://www.dropbox.com/s/x89iznixlfma3jh/TempleAnim2MantaflowProblem_2.blend?dl=0